Hi @Anonymous , welcome to Microsoft Q&A forum and sorry for the trouble you have been facing.
We should be able to start/stop the service without any issue. I tried it do it through Azure Portal as well as below CLI commands and was able to do that.
az mysql flexible-server stop --name <servername> --resource-group <resourcegrpname>
az mysql flexible-server start --name <servername> --resource-group <resourcegrpname>
Could you please also try to run these CLI commands and check if they work for you? I tried it in East US region. Which region are you using?